Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,11)(7,13)(9,19)(14,15)(16,20)(17,18);(5,6)(7,12)(8,16)(9,14)(10,15)(11,19)(13,17)(18,20);(5,7,9,13,10,16,19,20)(6,12,14,17,15,8,11,18);(5,8)(6,13)(7,11)(9,18)(10,12)(14,16)(15,20)(17,19)Polycyclic presentation:
2^8|a^2=g,c^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136327, #136361, #136362, … |
| maximal normal subgroup of | #136327, #136361, #136362, … |
| maximal quotient of | #225494, #225499, #225504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15677);